Using the GIT approach of King [1]} to constructing moduli of quiver representations the \(G\) -Hilbert scheme can also be realised as a moduli space of \(\theta \) -stable quiver representations \(\mathcal {M}_\theta (Q,{d})\) , where \(Q\)  is the McKay quiver of \(G\)  and \({d}=(d_i)\)  is a given dimension vector. In this situation the stability parameter \(\theta \)  lives in the stability space
\(\Theta :=\Bigg \lbrace \theta \in \operatorname{Hom}_\mathbb {Z}\big (\mathbb {Z}^{Q_0},\mathbb {Q}\big )\colon \sum _{i\in Q_0} d_i\theta (i)=0\Bigg \rbrace ,\) 
